= Solution
For each fixed $\theta$, the <Taylor expansion> of the exponential yields
$$
\log M_{U_t}(\theta)=\frac{\theta^2}2+O((\lambda t)^{-1/2})\ \longrightarrow\ \frac{\theta^2}2.
$$
The limiting moment-generating <function> $e^{\theta^2/2}$ is that of the standard <normal distribution> and is finite around zero. The <continuity theorem for moment-generating functions>[moment-generating function continuity theorem] therefore gives
$$
\boxed{U_t\xrightarrow{d}N(0,1)}.
$$
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